Consider a random walk, say \(S_{n} = \xi _{1} + \dots + \xi _{n}\)  for every \(n \geqslant 1\) , where \((\xi _{i})_{i\geqslant 1}\)  is a sequence of i.i.d. copies of a random variable \(\xi \) .
The question of estimating quantities of the form \(\mathbb {P}(S_{n} \in (x,x+T])\)  for \(T \in (0,\infty ]\)  is fundamental in the study of random walks with numerous applications (e.g. in statistics, risk theory, statistical mechanics, queueing theory, etc.), and has received considerable interest, in particular in large deviation regimes. When the tail of the jump distribution vanishes exponentially fast (which is roughly speaking the so-called Cramér condition), seminal results go back to Cramér [1]} with extensions by Bahadur & Ranga Rao [2]} and Petrov [3]}; see also Höglund [4]}. The class of subexponential random walks (for which the Cramér condition does not hold), which bears close connections with the so-called `one-big jump prinicple', has also received considerable interest, see the important paper by Denisov, Dieker & Schneer [5]} and references therein. Quite some effort has also been devoted to the much more delicate estimation of `local' probabilities (corresponding to \(T<\infty \) ), and whole books have been devoted to the subject; see in particular Borovkov & Borovkov [6]} for an exhaustive account.
